Displaying similar documents to “A note on local automorphisms”

On local automorphisms and mappings that preserve idempotents

Matej Brešar, Peter Šemrl (1995)

Studia Mathematica

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Let B(H) be the algebra of all bounded linear operators on a Hilbert space H. Automorphisms and antiautomorphisms are the only bijective linear mappings θ of B(H) with the property that θ(P) is an idempotent whenever P ∈ B(H) is. In case H is separable and infinite-dimensional, every local automorphism of B(H) is an automorphism.

The set of automorphisms of B(H) is topologically reflexive in B(B(H))

Lajos Molnár (1997)

Studia Mathematica

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The aim of this paper is to prove the statement announced in the title which can be reformulated in the following way. Let H be a separable infinite-dimensional Hilbert space and let Φ: B(H) → B(H) be a continuous linear mapping with the property that for every A ∈ B(H) there exists a sequence ( Φ n ) of automorphisms of B(H) (depending on A) such that Φ ( A ) = l i m n Φ n ( A ) . Then Φ is an automorphism. Moreover, a similar statement holds for the set of all surjective isometries of B(H).

Multiplicative maps that are close to an automorphism on algebras of linear transformations

L. W. Marcoux, H. Radjavi, A. R. Sourour (2013)

Studia Mathematica

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Let be a complex, separable Hilbert space of finite or infinite dimension, and let ℬ() be the algebra of all bounded operators on . It is shown that if φ: ℬ() → ℬ() is a multiplicative map(not assumed linear) and if φ is sufficiently close to a linear automorphism of ℬ() in some uniform sense, then it is actually an automorphism; as such, there is an invertible operator S in ℬ() such that φ ( A ) = S - 1 A S for all A in ℬ(). When is finite-dimensional, similar results are obtained with the mere assumption...

Maps on idempotent operators

Peter Šemrl (2007)

Banach Center Publications

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The set of all bounded linear idempotent operators on a Banach space X is a poset with the partial order defined by P ≤ Q if PQ = QP = P. Another natural relation on the set of idempotent operators is the orthogonality relation defined by P ⊥ Q ⇔ PQ = QP = 0. We briefly survey known theorems on maps on idempotents preserving order or orthogonality. We discuss some related results and open problems. The connections with physics, geometry, theory of automorphisms, and linear preserver...

Two characterizations of automorphisms on B(X)

Peter Šemrl (1993)

Studia Mathematica

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Let X be an infinite-dimensional Banach space, and let ϕ be a surjective linear map on B(X) with ϕ(I) = I. If ϕ preserves injective operators in both directions then ϕ is an automorphism of the algebra B(X). If X is a Hilbert space, then ϕ is an automorphism of B(X) if and only if it preserves surjective operators in both directions.

Locally spectrally bounded linear maps

M. Bendaoud, M. Sarih (2011)

Mathematica Bohemica

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Let ℒ ( ℋ ) be the algebra of all bounded linear operators on a complex Hilbert space ℋ . We characterize locally spectrally bounded linear maps from ℒ ( ℋ ) onto itself. As a consequence, we describe linear maps from ℒ ( ℋ ) onto itself that compress the local spectrum.

Notes on automorphisms of ultrapowers of II₁ factors

David Sherman (2009)

Studia Mathematica

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In functional analysis, approximative properties of an object become precise in its ultrapower. We discuss this idea and its consequences for automorphisms of II₁ factors. Here are some sample results: (1) an automorphism is approximately inner if and only if its ultrapower is ℵ₀-locally inner; (2) the ultrapower of an outer automorphism is always outer; (3) for unital *-homomorphisms from a separable nuclear C*-algebra into an ultrapower of a II₁ factor, equality of the induced traces...